Obj. 16 Trigonometric Functions (Presentation)

8/3/2019 Obj. 16 Trigonometric Functions (Presentation)

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Obj. 16 Trigonometric Functions

Unit 5 Trigonometric and Circular Functions


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Concepts and Objectives

Definitions of Trigonometric and Circular Functions

(Obj. #16) Find the values of the six trigonometric functions of

angle .

Find the function values of quadrantal angles.

Identify the quadrant of a given angle.

Find the other function values given one value and

the quadrant


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Trigonometric Ratio Review

In Geometry, we learned that for any given right triangle,

there are special ratios between the sides.

A

opposite

adjacent

=opposite

sin

hypotenuse

A

=adjacent

coshypotenuse

A

=opposite

tanadjacent

A


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Trigonometric Functions

Consider a circle centered at the origin with radius r:

The equation for this circle isx2 +y2 = r2

A point(x,y) on the circle creates a right triangle whose

sides arex,y, and r.

The trig ratios are now (x,y)r

x

y

=siny

r

=cos

x

r

=tany


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Trigonometric Functions

There are three other ratios in addition to the three we

already know : cosecant, secant, and cotangent. These ratios are the inverses of the original three:

(x,y)r

x

y

= =1

csc sin

r

y

= =1

seccos

r

x

= =

1cot tan

x

y


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Finding Function Values

Example: The terminal side of an angle in standard

position passes through the point(15, 8). Find thevalues of the six trigonometric functions of angle .

8

15

(15, 8)

We know thatx= 15 andy= 8, but

we still have to calculate r:

Now, we can calculate the values.

= +2 2

r x y

= + =2 2

15 8 1717


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Finding Function Values

Example: The terminal side of an angle in standard

position passes through the point(15, 8). Find thevalues of the six trigonometric functions of angle .

8

15

(15, 8)

17

= =8

sin

17

y

r

= =15

cos17

x

r

= =

8tan 15

y

x

= =17

csc

8

r

y

= =17

sec15

r

x

= =

15cot 8

x

y


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The Unit Circle

Angles in standard position whose terminal sides lie on

thex-axis ory-axis (90, 180, 270, etc.) are calledquadrantal angles.

To find function values of quandrantal angles easily, we

Notice what the differentx,y,

and rvalues are at the quadrantal

angle points (xandyare either 0,1, or 1; ris always 1).

use a circle with a radius of 1, which

is called a unit circle.


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Values of Quadrantal Angles

Example: Find the values of the six trigonometric

functions for an angle of 270.At 270,x= 0,y= 1, r= 1.

= =

1

sin270 11

= =0

cos270 01

= =

1tan270 undefined

0


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Values of Quadrantal Angles

Example: Find the values of the six trigonometric

functions for an angle of 270.At 270,x= 0,y= 1, r= 1.

= =

1

csc270 11

= =1

sec270 undefined0

= =

0cot 270 0

1


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Identifying an Angles Quadrant

To identify the quadrant of an angle given certain

conditions, note the following: In the first quadrant,xandyare both positive.

In QII,xis negative andyis positive.

In QIII, both are negative. In QIV,xis positive andyis

IVIII

II I

(+,+)(,+)

(,)

negative.

(+,)


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Identifying an Angles Quadrant

Example: Identify the quadrant (or possible quadrants)

of an angle that satisfies the given conditions.

a) sin > 0, tan < 0 b) cos < 0, sec < 0

I, II II, IV

II

II, III II, III

II, III


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Homework

College Algebra

Page 512: 30-78 (3s, omit 63), 93-102 (3s) HW: 42, 54, 72, 78, 96, 102

Classwork: Algebra & Trigonometry(green book) Page 728: 77-78

Obj. 16 Trigonometric Functions (Presentation)

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